Problem 20 - Precalculus Mathematics for Calculus

Question

Express the given vector in terms of the unit vectors i, \(\mathbf{j}\). and \(\mathbf{k}\). $$\langle 0,-3,5\rangle$$

Step-by-Step Solution

Answer

\( 0\mathbf{i} - 3\mathbf{j} + 5\mathbf{k} \)

Step 1: Identify the Components

The vector is given as \( \langle 0, -3, 5 \rangle \). Note that there are three components: 0, -3, and 5. These components correspond to the \( x \), \( y \), and \( z \) directions, respectively.

Step 2: Break Down the Vector

Write the vector as a sum of its components multiplied by the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \). Each unit vector corresponds to one of the axes: \( \mathbf{i} \) for \( x \), \( \mathbf{j} \) for \( y \), and \( \mathbf{k} \) for \( z \).

Step 3: Formulate the Expression

Combine the components with their respective unit vectors: \( 0\mathbf{i} + (-3)\mathbf{j} + 5\mathbf{k} \). This separates the vector into its components along the \( x \), \( y \), and \( z \) directions.

Key Concepts

Understanding Unit Vectors

Unit vectors are essential building blocks in the world of vectors. They are vectors with a length or magnitude of one. By being direction indicators, unit vectors help us express more complex vectors in terms of standardized directions. Unit vectors in three-dimensional space align with the axes of a 3D coordinate system. These are commonly represented as:

• \( \mathbf{i} \) along the x-axis
• \( \mathbf{j} \) along the y-axis
• \( \mathbf{k} \) along the z-axis

Using unit vectors, any vector can be broken down into its components. This simplifies the understanding and handling of vectors because we're using a standardized framework where the directions are always consistent. Think of unit vectors as the cardinal directions of a compass in three-dimensional space, pointing exactly along the axes with no deviation.

Exploring Vector Components

Vector components break down a vector into its parts based on the directions of the axes in a coordinate system. For a vector like \( \langle 0, -3, 5 \rangle \), the numbers \( 0 \), \( -3 \), and \( 5 \) are its components. Each component tells us how far the vector stretches along each respective axis.

In general:

• The first component (0) stretches along the x-axis
• The second component (-3) stretches along the y-axis
• The third component (5) stretches along the z-axis

Why do we care about vector components? They help us understand the vector's direction and magnitude in terms of the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \). By writing \( 0\mathbf{i} + (-3)\mathbf{j} + 5\mathbf{k} \), you visualize a vector as a sum of its directional pulls. It illustrates clearly how the vector behaves in 3D space.

Understanding 3D Vectors

3D vectors represent quantities with both direction and magnitude in three-dimensional space. They are expressed as \( \langle x, y, z \rangle \), where \( x \), \( y \), and \( z \) correspond to the projection of the vector along the x, y, and z axes, respectively. Understanding these vectors helps in physics and engineering to convey information about forces, velocities, and more.To express a 3D vector using unit vectors, you use expressions such as \( x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \). This means you're aligning each component of the vector with its respective axis, providing clear directional sense and magnitude.

Visualizing a 3D vector involves seeing how much it "pushes" along each axis. Consider our example vector \( \langle 0, -3, 5 \rangle \), which signifies no movement along the x-axis, a movement backward (negative direction) along the y-axis, and a positive movement along the z-axis. Understanding and communicating these vectors in their component form allows us to accurately manipulate and calculate various real-world scenarios, as we can easily define the vector's influence in each direction.